A Generalized Faulhaber Inequality, Improved Bracketing Covers, and Applications to Discrepancy

TL;DR

This paper introduces a generalized Faulhaber inequality to improve bounds on bracketing numbers and applies these results to enhance understanding of discrepancy measures for random point sets in high-dimensional spaces.

Contribution

It presents a new generalized Faulhaber inequality and uses it to derive tighter bounds on bracketing numbers and discrepancy measures, advancing the theoretical understanding of these concepts.

Findings

Improved bounds for bracketing numbers of axis-parallel boxes.

New bounds for star-discrepancy of negatively dependent random point sets.

Enhanced estimates for weighted star-discrepancy.

Abstract

We prove a generalized Faulhaber inequality to bound the sums of the $j$-th powers of the first $n$ (possibly shifted) natural numbers. With the help of this inequality we are able to improve the known bounds for bracketing numbers of $d$-dimensional axis-parallel boxes anchored in $0$ (or, put differently, of lower left orthants intersected with the $d$-dimensional unit cube $[0,1]^d$). We use these improved bracketing numbers to establish new bounds for the star-discrepancy of negatively dependent random point sets and its expectation. We apply our findings also to the weighted star-discrepancy.